An Empirical Investigation of Direct and Iterated Multistep Conditional Forecasts Michael W. McCracken Joseph T. McGillicuddy Federal Reserve Bank Federal Reserve Bank of St. Louis 1 of St. Louis Deutsche Bundesbank September 9, 2017 1 The views expressed herein are solely those of the authors and do not necessarily reflect the views of the Federal Reserve Bank of St. Louis or the Federal Reserve System. McCracken and McGillicuddy Conditional DMS vs. IMS September 9, 2017 1 / 27
Introduction Central banks use conditional forecasts to assess hypothetical policies. Christoffel et al. (ECB, 2008) Large banks have to construct conditional forecasts as part of stress testing exercises. Sarychev (BoE, 2014) Recent surge in conditional forecasting in academic literature: � Giannone et al. (NY Fed, 2014): Big VARs � Baumeister and Kilian (BoC, 2013): Oil � Aastveit et al. (NB, 2014): Break tests � Clark and McCracken (Feds, 2017): Inference All of these are VARs or near-VARs using IMS approach McCracken and McGillicuddy Conditional DMS vs. IMS September 9, 2017 2 / 27
Does DMS vs. IMS Matter for Forecasting? For unconditional point forecasts, theory says “Yes.” � Bhansali (1997) � Schorfheide (2005) For unconditional point forecasts, empirics say “Yes.” � Marcellino, Stock, and Watson (MSW; 2006) For impulse response functions, empirics say “Yes.” � Jorda (2008) McCracken and McGillicuddy Conditional DMS vs. IMS September 9, 2017 3 / 27
What We Do We provide empirical evidence on whether the DMS vs IMS battle matters for conditional point forecasts. � IMS is MSE optimal from OLS VAR (Waggoner and Zha, 1999) � DMS is just OLS BLP (Goldberger, 1962) We do so by shamelessly emulating what MSW did. � 2000 bivariate systems used to construct MSEs � 150 trivariate “monetary” systems used to construct MSEs Scenarios are ex-post realized actuals so that scenarios actually occur and MSE is relevant. McCracken and McGillicuddy Conditional DMS vs. IMS September 9, 2017 4 / 27
Overview of Results Whether our results coincide with MSW is sample dependent � Over common sample, our results align with theirs: IMS generally favored with some benefits to DMS for nominals � For Great Moderation sample, DMS heavily favored for nominals and about the same as IMS for reals/financials Robustness � Bivariate and trivariate � horizons: h = 3 , 6 , 12 , 24 � lag selection: fixed at 4 or 12, or AIC / BIC � real-time vintage data McCracken and McGillicuddy Conditional DMS vs. IMS September 9, 2017 5 / 27
What We Don’t Do Consider bayesian estimation. Consider large VARs. Consider univariate ARDLs (Guerrieri and Welch 2014) or alternative DMS in Jorda and Marcellino (2008). McCracken and McGillicuddy Conditional DMS vs. IMS September 9, 2017 6 / 27
DMS vs. IMS examples McCracken and McGillicuddy Conditional DMS vs. IMS September 9, 2017 7 / 27
Examples of DMS vs. IMS VAR(1) taking the form � y t � � 0 � � y t − 1 � � e t � b = + , x t 0 c x t − 1 v t with i.i.d. N(0,1) errors with contemporaneous correlation ρ . Use pseudo-true parameters used for forecasting One-step conditional forecast of y t + 1 given x t + 1 DMS model is trivial: y t = γ x t + η t McCracken and McGillicuddy Conditional DMS vs. IMS September 9, 2017 8 / 27
Correct Specification y c IMS forecast is ˆ t , 1 = bx t + ρ ( x t + 1 − cx t ) t , 1 ) 2 = 1 − ρ 2 . � E ( e IMS y c t , 1 = ( bc + ρ ( 1 − c 2 )) x t + 1 DMS forecast is ˆ ) 2 = 1 − ρ 2 + ( b − ρ c ) 2 . � E ( e DMS t , 1 Minimum-MSE IMS better. McCracken and McGillicuddy Conditional DMS vs. IMS September 9, 2017 9 / 27
VAR Misspecification of Conditional Mean Equation for x misspecified as x t = α x t − 2 + η t ρ y c 1 + c 2 ( x t + 1 − c 2 x t ) IMS forecast is ˆ t , 1 = bx t + √ 2 ρ 2 t , 1 ) 2 = 1 + ρ 2 − � E ( e IMS √ 1 + c 2 y c t , 1 = ( bc + ρ ( 1 − c 2 )) x t + 1 DMS forecast is ˆ ) 2 = 1 − ρ 2 + ( b − ρ c ) 2 . � E ( e DMS t , 1 DMS better if b = ρ 2 and c = ρ . McCracken and McGillicuddy Conditional DMS vs. IMS September 9, 2017 10 / 27
VAR Misspecification of Residual Variance Conditional mean ok. Residual correlation changes from ρ 0 to ρ 1 between T and T + 1 . y c T , 1 = bx T + ρ 0 ( x T + 1 − cx T ) IMS forecast is ˆ T , 1 ) 2 = 1 + ρ 2 � E ( e IMS 0 − 2 ρ 0 ρ 1 y c T , 1 = ( bc + ρ 0 ( 1 − c 2 )) x T + 1 DMS forecast is ˆ T , 1 ) 2 = 1 + b 2 + ρ 2 � E ( e DMS 0 ( 1 − c 2 ) − 2 bc ρ 1 − 2 ρ 0 ρ 1 ( 1 − c 2 ) . DMS better if b = 0 and c ∈ ( 0 , 1 ) , ρ 0 ∈ ( − 1 , 0 ) , and ρ 1 ∈ ( 0 , 1 ) . McCracken and McGillicuddy Conditional DMS vs. IMS September 9, 2017 11 / 27
Methodology McCracken and McGillicuddy Conditional DMS vs. IMS September 9, 2017 12 / 27
VAR-based IMS Conditional Forecasts Z t = ( Y t , X t ) � in levels or log-levels. Let z t = ( y t , x t ) � denote the stationary transformation: level or differences. OLS estimate of VAR for z t = ( y t , x t ) � : z t = C + A ( L ) z t − 1 + e t used to produce h -step conditional forecasts of the form y c y u x u ˆ t , h = ˆ t , h + ∑ 1 ≤ i ≤ h ˆ γ i , t ( x t + i − ˆ t , i ) γ i , t that are known functions of ˆ A i , t , and ˆ for constants ˆ Σ t . y c ˆ if Y t is I ( 0 ) t , h Y c ˆ Y t + ∑ h y c t , h = i = 1 ˆ if Y t is I ( 1 ) . t , i Y t + h ∆ Y t + ∑ h i = 1 ∑ i y c j = 1 ˆ if Y t is I ( 2 ) t , j McCracken and McGillicuddy Conditional DMS vs. IMS September 9, 2017 13 / 27
ARDL-based DMS Conditional Forecasts Define if Y t is I ( 0 ) Y t y ( h ) = Y t − Y t − h if Y t is I ( 1 ) . t Y t − Y t − h − h ∆ Y t if Y t is I ( 2 ) OLS estimate of ARDL for z t = ( y t , x t ) � : y ( h ) = α + ∑ p − 1 j = 0 β j y t − h − j + ∑ p − 1 j = 0 δ j x t − h − j + ( ∑ 1 ≤ i ≤ h γ i x t − h + i ) + ε t t used to produce h -step conditional forecasts of the form y c ( h ) α t + ∑ p − 1 β j , t y t − j + ∑ p − 1 j = 0 ˆ j = 0 ˆ = ˆ δ j , t x t − j + ( ∑ 1 ≤ i ≤ h ˆ ˆ γ i , t x t + i ) t , h ˆ Y c t , h computed relative to order of integration of Y . McCracken and McGillicuddy Conditional DMS vs. IMS September 9, 2017 14 / 27
Choices Lag selection: 4, 12, or updated AIC/BIC Horizons: 3, 6, 12, 24 Paths: Future x known throughout forecast horizon In-sample: Start in 1959:01 or 1984:01 Out-of-sample: 1979:01+ h — 2002:12 or 2002:12+ h — 2016:12 McCracken and McGillicuddy Conditional DMS vs. IMS September 9, 2017 15 / 27
Bivariate Data Bivariate results use June 2017 vintage of 121 monthly series taken from FRED-MD (McCracken and Ng, 2016). � Series mapped into the 5 groups MSW have: a) Income, output, sales, and capacity utilization b) Employment and unemployment c) Construction, inventories, and orders d) Interest rates and asset prices e) Nominal prices, wages, and money � MSW have 170 series: differences mostly come from groups (a) and (c) above The 2,000 bivariate systems are obtained at random by selecting pairs ( y , x ) from distinct groups (200 from each of the 10 possible group pairs). From these, 4,000 forecasts made for each lag-selection method, horizon, and forecasting approach (IMS or DMS). McCracken and McGillicuddy Conditional DMS vs. IMS September 9, 2017 16 / 27
Trivariate Data For trivariate results we use 6 real, 5 nominal, and 5 financial series Real Nominal 1 real personal income 1 avg. hourly mfg. earnings 2 IP 2 PPI 3 employment 3 oil price 4 unemployment rate 4 CPI 5 avg. weekly mfg. hours 5 PCEPI 6 real personal consumption Financial 1 M1 money stock 2 fed funds rate 3 10-year treasury 4 trade-weighted US dollar index 5 S&P 500 Most results from June 2017 vintage but some from real-time data The 150 trivariate systems provide 450 forecasts for each lag-selection method, horizon, and forecasting approach (IMS or DMS). McCracken and McGillicuddy Conditional DMS vs. IMS September 9, 2017 17 / 27
Empirical Results McCracken and McGillicuddy Conditional DMS vs. IMS September 9, 2017 18 / 27
Bivariate Results: Conditional on Full Path, Sample Start 1959:01, Out-of-Sample Period 1979:01 + h to 2002:12 (A) All variables (B) Pairs excl. PWM (C) PWM variables Forecast Horizon: 3 12 24 3 12 24 3 12 24 AR(4) Mean 1.00 1.04 1.11 1.00 1.05 1.12 0.98 0.92 0.96 Median 1.00 1.02 1.07 1.01 1.04 1.09 0.98 0.91 0.94 IMS better 3.2% 7.5% 11.2% 3.5% 8.6% 12.8% 0.8% 0.0% 1.0% DMS better 5.7% 8.2% 5.9% 5.2% 4.3% 2.8% 10.5% 27.3% 19.6% AR(12) Mean 1.02 1.08 1.18 1.02 1.08 1.18 1.01 1.03 1.09 Median 1.01 1.06 1.13 1.01 1.06 1.13 1.01 1.03 1.06 IMS better 5.3% 10.4% 15.3% 4.8% 11.5% 16.7% 6.4% 6.4% 4.5% DMS better 0.9% 1.3% 2.0% 0.8% 0.9% 1.3% 1.6% 3.3% 5.3% BIC Mean 0.95 0.96 1.04 0.98 0.99 1.09 0.84 0.72 0.76 Median 0.99 0.98 1.04 0.99 0.99 1.06 0.83 0.70 0.73 IMS better 3.7% 6.3% 9.6% 4.5% 7.0% 11.6% 0.5% 0.1% 0.8% DMS better 17.7% 16.1% 11.3% 12.6% 8.3% 4.4% 46.3% 53.5% 41.0% McCracken and McGillicuddy Conditional DMS vs. IMS September 9, 2017 19 / 27
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